The Error Correctted And Exponential Stochastic Runge-Kutta Method

As an extremely important research direction in the field of mathematics, the stochastic differential equations hold a pivotal place in real life. Because of the complexity of the stochastic differential equation itself, the exact solutions of SDEs are hard to be obtained. This makes it more important to construct the numerical methods. As we all know, explicit numerical methods of stochastic differential equations are simple and computationally efficient, but most of the existing explicit numerical methods show poor stability, so they don’t work on stiff stochastic differential equations.This paper discusses two types of explicit modified numerical method based on an explicit stochastic Runge-Kutta method. These two types of numerical method not only retain good mean-square convergence of the original stochastic Runge-Kutta method with order 1, but also overcome the defect of poor stability. What’s more, the methods show pretty good stability.The first method called error corrected stochastic Runge-Kutta method(ECSRK) is achieved by adding the error correction term to an explicit stochastic Runge-Kutta method. It is proved that the mean-square convergent order of the numerical method is 1. Thereafter, the mean-square stability and asymptotic stability of ECSRK are discussed by applying the method to linear test equation. As a result, both the mean-square stability condition and asymptotic stability condition are obtained. And the mean-square stability region and asymptotic stability region are also plotted, respectively. Compared with the original stochastic Runge-Kutta method, the mean-square stability and asymptotic stability of ECSRK method are far better than that of Runge-Kutta method.Referred to exponential stochastic Runge-Kutta method, the second method is a generalization of the exponential Runge-Kutta method for solving ordinary differential equations. In this paper, error remainder of exponential stochastic Runge-Kutta method is analyzed by expanding exact solution of the stochastic differential equations via It? formula. The exponential stochastic Runge-Kutta method is turned out to be mean-square convergent with order 1. In terms of stability, this method is proved to be mean-square stable unconditionally. That is, the method preserves the mean-square stability without any constraint on the numerical step size. Furthermore, asymptotic stability domain of exponential stochastic Runge-Kutta method is much larger than that of the original Runge-Kutta method.All the theoretical results and conclusions are supported by corresponding numerical examples.